1217 Arbitrage

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本文介绍了一种使用Floyd算法检测货币兑换是否存在套利机会的方法。通过构建货币汇率图，利用Floyd算法更新每种货币之间的最短路径，检测是否存在从一种货币转换回自身且价值增加的情况，以此判断是否存在套利机会。

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#问题

Problem Description
Arbitrage is the use of discrepancies in currency exchange rates to transform one unit of a currency into more than one unit of the same currency. For example, suppose that 1 US Dollar buys 0.5 British pound, 1 British pound buys 10.0 French francs, and 1 French franc buys 0.21 US dollar. Then, by converting currencies, a clever trader can start with 1 US dollar and buy 0.5 * 10.0 * 0.21 = 1.05 US dollars, making a profit of 5 percent. 

Your job is to write a program that takes a list of currency exchange rates as input and then determines whether arbitrage is possible or not.
 

Input
The input file will contain one or more test cases. Om the first line of each test case there is an integer n (1<=n<=30), representing the number of different currencies. The next n lines each contain the name of one currency. Within a name no spaces will appear. The next line contains one integer m, representing the length of the table to follow. The last m lines each contain the name ci of a source currency, a real number rij which represents the exchange rate from ci to cj and a name cj of the destination currency. Exchanges which do not appear in the table are impossible.
Test cases are separated from each other by a blank line. Input is terminated by a value of zero (0) for n. 
 

Output
For each test case, print one line telling whether arbitrage is possible or not in the format "Case case: Yes" respectively "Case case: No". 
 

Sample Input
3
USDollar
BritishPound
FrenchFranc
3
USDollar 0.5 BritishPound
BritishPound 10.0 FrenchFranc
FrenchFranc 0.21 USDollar

3
USDollar
BritishPound
FrenchFranc
6
USDollar 0.5 BritishPound
USDollar 4.9 FrenchFranc
BritishPound 10.0 FrenchFranc
BritishPound 1.99 USDollar
FrenchFranc 0.09 BritishPound
FrenchFranc 0.19 USDollar

0
 

Sample Output
Case 1: Yes
Case 2: No

#AC码

/*
Floyd更新一遍，检查自己对自己的汇率是否大于1
*/
#include<bits/stdc++.h>
using namespace std;
#define Max_n 32
double D[Max_n][Max_n];
map<string,int> Map;
int N,M;
void Floyd()
{
    for(int k=1;k<=N;k++)
        for(int i=1;i<=N;i++)
            for(int j=1;j<=N;j++)
            {
                if(D[i][j]<D[i][k]*D[k][j])
                    D[i][j]=D[i][k]*D[k][j];
            }
}
int main()
{
    //freopen("1217.txt","r",stdin);
    int Case=0;
    while(cin>>N,N)
    {
        //
        Map.clear();
        for(int i=1;i<=N;i++)
            for(int j=1;j<=N;j++)
            {
                if(i=j)
                    D[i][j]=1;
                else
                    D[i][j]=0;
            }
        //
        for(int i=1;i<=N;i++)
        {
            string Name;
            cin>>Name;
            Map[Name]=i;
        }
        //
        cin>>M;
        for(int i=1;i<=M;i++)
        {
            string u,v;
            double cost;
            cin>>u>>cost>>v;
            D[Map[u]][Map[v]]=cost;
        }
        //
        Floyd();
        //
        int flag=0;
        for(int i=1;i<=N;i++)
        {
            if(D[i][i]>1)
            {
                flag=1;
                break;
            }
        }
        if(flag)
            cout<<"Case "<<++Case<<": Yes"<<endl;
        else
            cout<<"Case "<<++Case<<": No"<<endl;
    }
}